Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2412.17208

Local symmetries and extensive ground-state degeneracy of a 1D supersymmetric fermionic chain

Shuyu Zhang, Hiroki Sukeno, Kazuki Ikeda, Tzu-Chieh Wei

Published Dec 23, 2024• 0 citations
arXivHTMLPDF

Abstract

We study a $1$D supersymmetric (SUSY) hard-core fermion model first proposed by Fendley, Schoutens, and de Boer [Phys. Rev. Lett. 90, 120402 (2003)]. We focus on the full Hilbert space instead of a restricted subspace. Exact diagonalization shows the degeneracy of zero-energy states scales exponentially with size of the system, with a recurrence relation between different system sizes. We solve the degeneracy problem by showing the ground states can be systematically constructed by inserting immobile walls of fermions into the chain. Mapping the counting problem to a combinatorial one and obtaining the exact generating function, we prove the recurrence relation on both open and periodic chains. We also provide an explicit mapping between ground states, giving a combinatorial explanation of the recurrence relation.

DOI
10.48550/arxiv.2412.17208

Related papers

Order and supersymmetry at high filling zero-energy states on the triangular lattice

D. Galanakis, Christopher L. Henley, Stefanos Papanikolaou

We perform exact-diagonalization studies in $d=2$ dimensions for the Fendley and Schoutens model of hard-core and nearest-neighbor-excluding fermions that displays an exact nonrelativistic supersymmetry. Using clusters of all possible shapes up to 46 sites, we systematically study the behavior of the ground-state phase diagram as a function of filling. We focus on the highly degenerate zero-energy states found at fillings between $1/7$ and $\ensuremath{\sim}$1/5. At the lower end of that interval, at filling $1/7$, we explicitly show that the ground states are gapped crystals. Consistent with previous suggestions, we find that the extensive entropy of zero states peaks at a filling of $\ensuremath{\sim}$0.178. At the higher end of the interval, we find zero-energy ground states at fillings above $1/5$, contrary to previous expectations, which display nontrivial amplitude degeneracies.

arXiv:1206.2044•Physical Review B•Nov 5, 2012•9 citations

On the ground states of the $M_\ell$ models

Liza Huijse, Christian Hagendorf

We study the $M_\ell$ models for lattice fermions with supersymmetry introduced by Fendley, Nienhuis and Schoutens on one-dimensional chains. We determine the number of ground states as a function of the chain length as well as various boundary conditions by solving the corresponding cohomology problem. As an intermediate result we obtain the Cut (and Paste) cohomology isomorphism, which maps the cohomology problem of chains, whose lengths differ by $\ell+2$ sites, onto each other.

arXiv:1509.08879•Sep 29, 2015•6 citations

Complexity of Supersymmetric Systems and the Cohomology Problem

Chris Cade, P. Marcos Crichigno

We consider the complexity of the local Hamiltonian problem in the context of fermionic Hamiltonians with N=2 supersymmetry and show that the problem remains QMA -complete. Our main motivation for studying this is the well-known fact that the ground state energy of a supersymmetric system is exactly zero if and only if a certain cohomology group is nontrivial. This opens the door to bringing the tools of Hamiltonian complexity to study the computational complexity of a large number of algorithmic problems that arise in homological algebra, including problems in algebraic topology, algebraic geometry, and group theory. We take the first steps in this direction by introducing the k -local Cohomology problem and showing that it is QMA1 -hard and, for a large class of instances, is contained in QMA . We then consider the complexity of estimating normalized Betti numbers and show that this problem is hard for the quantum complexity class DQC1 , and for a large class of instances is contained in BQP . In light of these results, we argue that it is natural to frame many of these homological problems in terms of finding ground states of supersymmetric fermionic systems. As an illustration of this perspective we discuss in some detail the model of Fendley, Schoutens, and de Boer consisting of hard-core fermions on a graph, whose ground state structure encodes l -dimensional holes in the independence complex of the graph. This offers a new perspective on existing quantum algorithms for topological data analysis and suggests new ones.

arXiv:2107.00011•Quantum•Apr 30, 2024•17 citations

Interacting Kitaev chain with N=1 supersymmetry

Urei Miura, Kenji Shimomura, Keisuke Totsuka

Lattice models with supersymmetry are known to exhibit a variety of remarkable properties that do not exist in the relativistic models. In this paper, we introduce an interacting generalization of the Kitaev chain of Majorana fermions with $\mathcal{N}=1$ supersymmetry and investigate its low-energy properties, paying particular attention to the ground-state degeneracy and low-lying fermionic excitations. First, we establish the existence of a phase with spontaneously broken supersymmetry and a phase transition out of it with the help of variational arguments and the exact ground state. We then develop, based on the superfield formalism, a simple mean-field theory, in which the order parameters detect supersymmetry breaking, to understand the ground-state phases and low-lying Nambu-Goldstone fermions. At the solvable point (frustration-free point), the exact ground state of an open chain exhibits large degeneracy of the order of the system size, which is attributed to the existence of a zero-energy domain wall (dubbed kink or skink) separating the topological and trivial states of Majorana fermions. Our results may shed new light on the intriguing ground-state properties of supersymmetric lattice models.

Physical review. B./Physical review. B•Feb 26, 2024•5 citations

Zero-Energy Problems for Supersymmetric Hamiltonians on a Chain Are $\mathrm{QMA}_1$-Complete

Yibin Wang

We determine the computational complexity of deciding whether a supersymmetric quantum system on a one-dimensional chain has a zero-energy ground state. We prove $\mathrm{QMA}_1$-completeness for two input models, formulated in terms of Hermitian and nilpotent supercharges, respectively. In the Hermitian model, both the supercharge terms and the Hamiltonian have bounded range on the chain; in the nilpotent model, this geometric restriction applies only to the Hamiltonian. The hardness reduction from quantum satisfiability on a chain preserves exact zero energy and gives an inverse-polynomial ground-energy lower bound for negative instances. A common verification method establishes membership in $\mathrm{QMA}_1$ while allowing cancellations between supercharge terms.

quant-pharXiv:2608.29333•Sep 1, 2026•0 citations

Supersymmetry breaking and Nambu-Goldstone fermions in an extended Nicolai model

Noriaki Sannomiya, Hosho Katsura, Yu Nakayama

We study a model of interacting spinless fermions in a one dimensional lattice with supersymmetry (SUSY). The Hamiltonian is given by the anticommutator of two supercharges $Q$ and ${Q}^{\ifmmode\dagger\else\textdagger\fi{}}$, each of which is comprised solely of fermion operators and possesses one adjustable parameter $g$. When the parameter $g$ vanishes, the model is identical to the one studied by Nicolai [H. Nicolai, J. Phys. A 9, 1497 (1976)], where the zero-energy ground state is exponentially degenerate. On the other hand, in the large-$g$ limit the model reduces to the free-fermion chain with a four-fold degenerate ground state. We show that for finite chains SUSY is spontaneously broken when $g>0$. We also rigorously prove that for sufficiently large $g$ the ground-state energy density is nonvanishing in the infinite-volume limit. We further analyze the nature of the low-energy excitations by employing various techniques such as rigorous inequalities, exact numerical diagonalization, and renormalization group method with bosonization. The analysis reveals that the low-energy excitations are described by massless Dirac fermions (or Thirring fermions more generally), which can be thought of as Nambu-Goldstone fermions from the spontaneous SUSY breaking.

arXiv:1606.03947•Physical review. D/Physical review. D.•Aug 25, 2016•17 citations
We study a 111D supersymmetric (SUSY) hard-core fermion model first proposed by Fendley, Schoutens, and de Boer [Phys. Rev. Lett. 90, 120402 (2003)]. We focus on the full Hilbert space instead of a restricted subspace. Exact diagonalization shows the degeneracy of zero-energy states scales exponentially with size of the system, with a recurrence relation between different system sizes. We solve the degeneracy problem by showing the ground states can be systematically constructed by inserting immobile walls of fermions into the chain. Mapping the counting problem to a combinatorial one and obtaining the exact generating function, we prove the recurrence relation on both open and periodic chains. We also provide an explicit mapping between ground states, giving a combinatorial explanation of the recurrence relation.
We perform exact-diagonalization studies in d=2d=2d=2 dimensions for the Fendley and Schoutens model of hard-core and nearest-neighbor-excluding fermions that displays an exact nonrelativistic supersymmetry. Using clusters of all possible shapes up to 46 sites, we systematically study the behavior of the ground-state phase diagram as a function of filling. We focus on the highly degenerate zero-energy states found at fillings between 1/71/71/7 and $\ensuremath{\sim}$1/5. At the lower end of that interval, at filling 1/71/71/7, we explicitly show that the ground states are gapped crystals. Consistent with previous suggestions, we find that the extensive entropy of zero states peaks at a filling of $\ensuremath{\sim}$0.178. At the higher end of the interval, we find zero-energy ground states at fillings above 1/51/51/5, contrary to previous expectations, which display nontrivial amplitude degeneracies.
On the ground states of the MℓM_\ellMℓ​ models
We study the MℓM_\ellMℓ​ models for lattice fermions with supersymmetry introduced by Fendley, Nienhuis and Schoutens on one-dimensional chains. We determine the number of ground states as a function of the chain length as well as various boundary conditions by solving the corresponding cohomology problem. As an intermediate result we obtain the Cut (and Paste) cohomology isomorphism, which maps the cohomology problem of chains, whose lengths differ by ℓ+2\ell+2ℓ+2 sites, onto each other.
We consider the complexity of the local Hamiltonian problem in the context of fermionic Hamiltonians with N=2 supersymmetry and show that the problem remains QMA -complete. Our main motivation for studying this is the well-known fact that the ground state energy of a supersymmetric system is exactly zero if and only if a certain cohomology group is nontrivial. This opens the door to bringing the tools of Hamiltonian complexity to study the computational complexity of a large number of algorithmic problems that arise in homological algebra, including problems in algebraic topology, algebraic geometry, and group theory. We take the first steps in this direction by introducing the k -local Cohomology problem and showing that it is QMA1 -hard and, for a large class of instances, is contained in QMA . We then consider the complexity of estimating normalized Betti numbers and show that this problem is hard for the quantum complexity class DQC1 , and for a large class of instances is contained in BQP . In light of these results, we argue that it is natural to frame many of these homological problems in terms of finding ground states of supersymmetric fermionic systems. As an illustration of this perspective we discuss in some detail the model of Fendley, Schoutens, and de Boer consisting of hard-core fermions on a graph, whose ground state structure encodes l -dimensional holes in the independence complex of the graph. This offers a new perspective on existing quantum algorithms for topological data analysis and suggests new ones.
Lattice models with supersymmetry are known to exhibit a variety of remarkable properties that do not exist in the relativistic models. In this paper, we introduce an interacting generalization of the Kitaev chain of Majorana fermions with N=1\mathcal{N}=1N=1 supersymmetry and investigate its low-energy properties, paying particular attention to the ground-state degeneracy and low-lying fermionic excitations. First, we establish the existence of a phase with spontaneously broken supersymmetry and a phase transition out of it with the help of variational arguments and the exact ground state. We then develop, based on the superfield formalism, a simple mean-field theory, in which the order parameters detect supersymmetry breaking, to understand the ground-state phases and low-lying Nambu-Goldstone fermions. At the solvable point (frustration-free point), the exact ground state of an open chain exhibits large degeneracy of the order of the system size, which is attributed to the existence of a zero-energy domain wall (dubbed kink or skink) separating the topological and trivial states of Majorana fermions. Our results may shed new light on the intriguing ground-state properties of supersymmetric lattice models.
Zero-Energy Problems for Supersymmetric Hamiltonians on a Chain Are QMA1\mathrm{QMA}_1QMA1​-Complete
We determine the computational complexity of deciding whether a supersymmetric quantum system on a one-dimensional chain has a zero-energy ground state. We prove QMA1\mathrm{QMA}_1QMA1​-completeness for two input models, formulated in terms of Hermitian and nilpotent supercharges, respectively. In the Hermitian model, both the supercharge terms and the Hamiltonian have bounded range on the chain; in the nilpotent model, this geometric restriction applies only to the Hamiltonian. The hardness reduction from quantum satisfiability on a chain preserves exact zero energy and gives an inverse-polynomial ground-energy lower bound for negative instances. A common verification method establishes membership in QMA1\mathrm{QMA}_1QMA1​ while allowing cancellations between supercharge terms.
We study a model of interacting spinless fermions in a one dimensional lattice with supersymmetry (SUSY). The Hamiltonian is given by the anticommutator of two supercharges QQQ and ${Q}^{\ifmmode\dagger\else\textdagger\fi{}}$, each of which is comprised solely of fermion operators and possesses one adjustable parameter ggg. When the parameter ggg vanishes, the model is identical to the one studied by Nicolai [H. Nicolai, J. Phys. A 9, 1497 (1976)], where the zero-energy ground state is exponentially degenerate. On the other hand, in the large-ggg limit the model reduces to the free-fermion chain with a four-fold degenerate ground state. We show that for finite chains SUSY is spontaneously broken when g>0g>0g>0. We also rigorously prove that for sufficiently large ggg the ground-state energy density is nonvanishing in the infinite-volume limit. We further analyze the nature of the low-energy excitations by employing various techniques such as rigorous inequalities, exact numerical diagonalization, and renormalization group method with bosonization. The analysis reveals that the low-energy excitations are described by massless Dirac fermions (or Thirring fermions more generally), which can be thought of as Nambu-Goldstone fermions from the spontaneous SUSY breaking.